So what would happen during that slow walking, would the troops be forcibly injected?
"Forcibly injected" like held down against your will? No, nothing like that. What happens is you get told to be at X location for vaccination. You don't want the jab? You don't show up or you do show up and try to get them to stamp your paper without the actual jab. If you cause a scene they probably just ask you to leave.
Now your name might show up on a spreadsheet as noncompliant, and since you didn't obey a valid order to show up at X and get the vaccine, you can now be punished under the UCMJ for at least one article (the military word for a "crime", some of which aren't crimes at all). Usually this would be an article 92, but when the command wants to throw the book at someone, they'll throw on multiple articles just to be petty, and because the punishment is decided and carried out by the command, it's up to them to be as light or as severe as they want to be.
If you're a twenty year E-7 putting in your retirement, rarely would you get rank taken on the way out (and your final rank determines your retirement benefits), but it's possible, and a very effective motivator for guys on the way out because they've been making plans for years on what to do when they get out based on that check being the size they think is coming to them.
If you're twenty years into your military career you are also used to just giving the military what it wants, so you probably just take the jab and hope whatever screws you up is covered by the VA. The bulk of the refuseniks would be junior enlisted who haven't had their spirit totally broken. They're always the bulk of UCMJ articles because you don't get to 20 years of service by continually breaking rules, break enough and you get the big chicken dinner (bad conduct discharge), or you don't get an option to reenlist.